given that I have a normal that is not facing true to any axis , how can I change it to face in the direction of the axis closest to it ?

I want to let a normal face directly upward if the normal is closest to the upwards axis , and I want it to face directly left or right or forward or backwards if it closer to one of those direction , how would I do such ?

or I guess the question could be thrased as , how to determine in which direction a normal faces ?